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Bank of Canada Banque du Canada
Working Paper 2004-20 / Document de travail 2004-20
Commodity-Linked Bonds: A Potential Meansfor Less-Developed Countries
to Raise Foreign Capital
by
Joseph Atta-Mensah
ISSN 1192-5434
Printed in Canada on recycled paper
Bank of Canada Working Paper 2004-20
June 2004
Commodity-Linked Bonds: A Potential Meansfor Less-Developed Countries
to Raise Foreign Capital
by
Joseph Atta-Mensah
Monetary and Financial Analysis DepartmentBank of Canada
Ottawa, Ontario, Canada K1A [email protected]
The views expressed in this paper are those of the author.No responsibility for them should be attributed to the Bank of Canada.
iii
Contents
Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAbstract/Résumé. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Experiences with Commodity-Linked Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Gold-linked bonds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Silver-linked bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Oil-linked bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
2.4 Other forms of commodity-indexed securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. Ways to Protect Export Commodities from Price Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1 International commodity agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Futures market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
3.3 Countertrade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4 The Baker plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
3.5 Research on the policy of debt relief for LDCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4. A Model of Optimal External Debt Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1 Conventional debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Commodity-linked bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Net foreign debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 The government’s maximization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5 Optimal allocation of external debt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iv
Acknowledgements
This paper was started during the author’s visit to the Bank of Ghana, June–August 2002. The
author acknowledges the useful comments of Pierre St-Amant, Christian Calmès, Céline
Gauthier, and Scott Gusba. The author takes the opportunity to thank Dr. Paul Acquah (Governor,
Bank of Ghana) and Dr. Mahamudu Bawumia (Bank of Ghana) for the invitation to the Bank of
Ghana. The editorial advice of Glen Keenleyside is greatly appreciated. However, any errors or
omissions must be attributed to the author.
v
-
n
tunity
e
that
r
he
of the
f the
nt of
debt
ptible
s de ce
e se
ntrent
s sont
lon la
e à
s
ents
n
ys en
s
inter-
Abstract
The author suggests that commodity-linked bonds could provide a potential means for less
developed countries (LDCs) to raise money on the international capital markets, rather tha
through standard forms of financing. The issue of this type of bond could provide an oppor
for commodity-producing LDCs to hedge against fluctuations in their export earnings. The
author’s results show that the value of a commodity-linked bond increases as the price of th
commodity indexed to the bond rises; this suggests that, if LDCs had issued debt contracts
were tied to their main export commodities, then their debt load would decline along with
plummeting export prices (or export revenues). A simple portfolio rule derived by the autho
suggests that LDCs should issue more commodity-linked bonds than conventional debt if t
variance of the portfolio is greater than twice the spread between the expected total return
conventional debt and the commodity-linked bond. This rule supports the view that, if more o
LDCs’ debt were issued in the form of commodity-linked bonds, then the debt-service payme
the LDCs would decline along with export prices (or export revenues), thus lightening their
load.
JEL classification: F30, F34, F49, G13, G11, O16Bank classification: Development economics; Financial markets; International topics
Résumé
L’auteur voit dans les obligations indexées sur les prix des produits de base un levier susce
d’aider les pays en développement à se procurer des capitaux sur les marchés financiers
internationaux, de préférence aux méthodes classiques de financement. L’émission de titre
genre pourrait offrir à ceux de ces pays qui sont riches en matières premières un moyen d
prémunir contre les fluctuations de leurs recettes d’exportation. Les résultats de l’étude mo
que la valeur de ces obligations augmente avec le cours du produit de base sur lequel elle
indexées. Cela donne à penser que, si les pays en développement émettaient des contrats
d’emprunt référencés sur leurs principaux produits d’exportation, le fardeau de leur dette
s’allégerait quand les cours de ces produits (ou leurs recettes d’exportation) diminuent. Se
règle simple que propose l’auteur, les pays en développement devraient recourir davantag
l’émission d’obligations indexées sur les prix des matières premières qu’à celle d’obligation
ordinaires si la variance de leur dette est deux fois plus élevée que l’écart entre les rendem
totaux espérés des deux types d’obligations. Cette règle tend à confirmer les bienfaits qu’u
recours accru aux émissions d’obligations indexées aurait sur le fardeau de la dette des pa
développement, du fait que l’évolution du service de la dette suivrait alors celle des prix de
produits exportés (et des recettes correspondantes).
Classification JEL: F30, F34, F49, G13, G11, O16Classification de la Banque: Économie du développement; Marchés financiers; Questions nationales
1
debt,
s and
olatile
s to
s to
mic
rough
s of
rest)
a
ty or
f the
wo
inty,
and
ised
ult
tial
ed to
ve to
to the
ts of
ult.
, the
f a
nownd bond
1. Introduction
Less-developed countries (LDCs) have for years been faced with colossal foreign debt. This
which is denominated in U.S. dollars at floating interest rates, became impaired in the 1970
1980s when interest rates were very high. Moreover, unfavourable terms of trade, due to v
prices of export commodities and falling export revenue, have hampered the ability of LDC
retire and/or service their debts. Consequently, the debt “overhang” has limited their acces
new foreign capital, forcing them to adjust their domestic investment and consumption.
Unfortunately, the LDCs are still mired in a debt crisis, which is seriously stifling their econo
growth.
The purpose of this paper is to examine whether commodity-linked bonds could provide a
potential means for LDCs to raise money on the international capital markets, rather than th
standard forms of financing. Commodity-linked bonds differ from conventional bonds in term
their payoffs to the holder. The bearer of the conventional bond receives fixed coupon (inte
payments during the life of the bond, and face value (principal) at maturity. The principal of
commodity-linked bond, however, is paid in either the physical units of a reference commodi
its equivalent monetary value. Similarly, the coupon payments may or may not be in units o
commodity to which the bond is indexed. Therefore, the structural difference between the t
bonds is that the nominal return of the conventional bond held to maturity is known with certa
although the real return is unknown due to inflation uncertainty, whereas both the nominal
real returns of the commodity-linked bond are not known.
In both the conventional and the commodity-linked bonds, the payments referred to are prom
(or contractual). If the issuer is unable or unwilling to make the contractual payments, defa
occurs, and the bearer receives a smaller or zero payment. In the event of default, substan
bankruptcy, legal, and renegotiating costs may be incurred, and new uncertainties may be
introduced (especially in international borrowing). These are dead-weight losses (as oppos
simple wealth transfer) to the parties involved in the contract. Derivative securities may ser
minimize these dead-weight losses, in that the state-contingent payments may be tailored
risk preferences of either borrower or lender. This tailoring would avoid the transaction cos
using other markets for the same purpose, and would also minimize the probability of defa
There are two types of commodity-indexed bonds: forward and option. With the forward type
coupon and/or principal payment to the bearer of the bond are linearly related to the price o
stated amount of the reference commodity.1 With the option type, the coupon payments are
1. Technically, the forward type is known as the commodity-indexed bond, and the option type is kas the commodity-linked bond. Unless otherwise stated, however, the terms commodity-indexeand commodity-linked bond are used interchangeably.
2
s an
vely,
face
et
the
cond,
of an
d
ses as
debt
value
he
s in
g
is
ter of
ue of
h
), who
es
l of
bonds
form
similar to that of a conventional bond, but at maturity the bearer receives the face value plu
option to buy or sell a predetermined quantity of the commodity at a specified price. Alternati
to minimize the default risk, the borrower may be given the option to pay the minimum of the
value and the value of the reference amount of the commodity at the maturity date.
In this paper, two approaches are taken to examine the potential benefits of LDCs issuing
commodity-linked bonds. First, the theory of option pricing is applied to determine the mark
price of a commodity-linked bond. An assessment is then made as to whether the value of
commodity-linked bond decreases with the decrease in the underlying commodity price. Se
the model of Myers and Thompson (1989) is extended to determine the optimal proportion
LDC’s total external debt that must be issued by the country in the form of commodity-linke
bonds. The relationship between the commodity price and the demand for the bond is also
determined.
The results reported in this paper show that the value of the commodity-linked bond increa
the price of the commodity indexed to the bond rises, which suggests that if LDCs had issued
contracts that were tied to their main export commodities, then their debt load would have
declined along with plummeting export prices (or export revenues).
It is also demonstrated in this paper that the coupon rate for a conventional debt with a face
identical to that of a commodity-linked bond is generally less than the coupon rate for a
commodity-linked bond that pays holders, on maturity, the minimum of the face value and t
monetary value of a pre-specified unit of a commodity. This implies that LDCs or corporation
need of investment funds could share the appreciation of the market value of the underlyin
commodity with the bondholders, in return for a lower coupon rate.
The results reported in this paper also show that the coupon rate for the conventional bond
greater than its counterpart for a commodity-linked bond whose terminal payoff is the grea
the face value and the monetary value of a pre-specified unit of a commodity. Through the iss
such a bond, an LDC could share the depreciation of the market value of its commodity wit
bondholders in exchange for higher coupon rates. This result corroborates Caballero (2003
argues that bonds of this nature act as a hedge for LDCs in times when the commodity pric
collapse.
A simple portfolio rule a country could follow in its allocation of debt instruments and the leve
imports is also derived. The rule suggests that LDCs should issue more commodity-linked
than conventional debt. It supports the view that, if more of LDCs’ debts were issued in the
3
h
ces
to
n by
A
ted in
ate
lue
rity
he
ence
80, the
stead
ed at
onds
ry
eive
ed to
of commodity-linked bonds, the debt-service payment of the LDCs would decline along wit
export prices (or export revenues), thus lightening the debt load of the LDCs.
This paper is organized as follows. Section 2 provides a brief background on past experien
with the issue of commodity-linked bonds. Section 3 discusses avenues available to LDCs
protect their export commodity prices. Section 4 constructs a model of external debt allocatio
an LDC. Section 5 offers some conclusions.
2. Experiences with Commodity-Linked Bonds
In this section, previous experiences with commodity-linked bonds are summarized.2
2.1 Gold-linked bonds
The most popular form of commodity-indexed bond is referenced to specified units of gold.
well-known example of gold bonds was issued in 1973 by the French government and accep
the financial markets as the “Giscard.” The “Giscard” carried a 7 per cent nominal coupon r
and a redemption value indexed to the price of a 1 kilogram bar of gold. The bearers of the
“Giscard” were protected by a safeguard clause, which stated that interest and the face-va
payments would be indexed to a 1 kilogram bar of gold should the French franc lose its pa
with gold and other currencies. In 1977, to the disappointment of the French government, t
French franc was forced by other European currencies to float. Furthermore, in 1978, the
International Monetary Fund (IMF) abolished the linkage of currencies to gold. As a consequ
of these two economic events, the safeguard clause became operative and therefore, in 19
government of France paid 393 francs in interest payments for every single bond issued, in
of the 70 francs originally planned for. Moreover, each of the issued bonds, which was trad
par in 1977, matured in January 1988 with a redemption value of 8,910 francs. Thus, the b
increased in value by about 700 per cent over 10 years.
After the “Giscard,” other types of gold-linked securities were issued. Unlike the “Giscard,”
which had only its redemption value indexed to a specified amount of gold, they had their
principal and/or interest payments indexed to gold. One type was issued in 1981 by the
Refinement International Company: the gold bonds were 3.29 per cent gold-linked, with an
aggregate principal of 100,000 ounces of gold. The maturity date for the bonds was Februa
1996. Interest payments were made annually. Bearers of these bonds had the option to rec
both interest and principal in either the monetary value of the specified amount of gold index
2. This section has been influenced by Fall (1986) and Privolos and Duncan (1991).
4
ade
gold-
were
en
986,
rants
f
per
um of
was
h
to
f
that
e an
ces.
of
ancial
re S.A.
was
the bond, or the physical quantity of gold referenced. Claims for the units of gold could be m
in London or Zurich.
The gold warrants issued by Echo Bay Mines Ltd. of Canada in 1981 were another type of
indexed securities: they issued 1,550,000 preferred voting shares. Holders of these shares
entitled to an annual dividend of US$3 and four gold warrants per share. Each warrant, wh
exercised, guaranteed the holder 0.0706 ounces of gold from Echo Bay Mines at a price of
US$595 per ounce. The four warrants had to be exercised on different dates: 31 January 1
31 January 1987, 31 January 1988, and 31 January 1989, respectively. Holders of the war
were allowed to trade them to a third party before 30 December 1983. The exercise of the
warrants was dependent on the completion of the Lupin Gold project.
2.2 Silver-linked bonds
In 1980, the Sunshine Mining Company, a large silver mine in the United States, issued
US$25 million worth of silver-indexed bonds to hedge against the fluctuations in the price o
silver. Each US$1,000 bond was indexed to 50 ounces of silver, payed a coupon rate of 8.5
cent, and had a maturity of 15 years. At each bond’s maturity, its bearer received the maxim
the face value of US$1,000 or the market value of 50 ounces of silver. The bonds were
redeemable on or after 15 April 1995 only if the average silver price for 30 consecutive days
above US$40 per ounce.
Silver-indexed bonds were also issued by the Sunshine Mining company in April 1985. Eac
US$1,000 bond was referenced to 58 ounces of silver and the coupon rate was increased
9.75 per cent. On the maturity date of April 2004, the holders of the bonds had the option o
choosing the face value of US$1,000 or the market value of 58 ounces of silver.
Unlike the gold bonds, there are not many silver-linked securities, for the economic reason
the market price of silver has not fluctuated very much. Hence, silver producers do not hav
incentive to issue silver bonds for the sole purpose of hedging against changes in silver pri
2.3 Oil-linked bonds
Oil-backed bonds appeared in the financial market during the late 1970s. The government
Mexico is believed to have been the first to issue such bonds. These bonds, known in the fin
markets as Petrobonds, were issued on behalf of the government by the National Financie
(NAFINSA), a development bank owned by the Mexican government. Each 1,000 peso bond
linked to 1.95354 barrels of oil.
5
On the
alue or
e
, but
nd,
s.
1986)
d a
, was
crude
e of
y
vestor
the
s, the
(1991)
nd
nds
te of
te the
led
nt on
erage
e.
1988.
erest
and
The coupon rate was 12.65823 per cent per annum and matured at the end of three years.
maturity date, the Petrobonds were redeemed at a value equal to the maximum of the face v
the market value of the referenced units of oil plus all coupons received during the life of th
bond. With this issue, the government was not only raising new money at low nominal cost
was also hedging part of its oil production against fluctuations in oil prices. On the other ha
bearers of the Petrobonds were hoping to benefit from an upswing in the price of crude oil.
In 1981, Petro-Lewis Corporation of Denver issued US$20 million worth of oil-indexed note
Each note had a lifetime of five years and paid an annual coupon rate of 9 per cent. As Fall (
explains, each note was expected to pay the face value (principal), the accrued interest, an
contingent interest on the maturity date. The contingent interest, which had a feature of a cap
defined as the increase over US$668.96 of (i) the average crude oil price of 18.5 barrels of
oil for the three months ending 28 February 1986 or, (ii) if greater, the highest average pric
18.5 per cent barrels of crude oil, up to a maximum of US$1,258 or US$68 per barrel for an
calendar quarter through the quarter ending 31 December 1985. This feature enabled an in
to make at most an additional US$589 per bond. The oil notes of Petro-Lewis differed from
Petrobonds in that the repayment of the face value included a call option on oil prices, and
therefore offered protection to the bearers from a fall in oil prices. In the case of Petrobond
payment of the principal was fully indexed to specified units of oil.
2.4 Other forms of commodity-indexed securities
Other bonds have been indexed to other types of precious metals. As Privolos and Duncan
report, Inco, which is one of the world’s largest producers of nickel, copper, silver, cobalt, a
platinum, in 1984 raised Can$90 million on the financial market through the issuance of bo
linked to the price of nickel or copper. The bonds, which matured in 1991, paid a coupon ra
10 per cent per annum. Holders of the bonds had the option of receiving at the maturity da
face value or the monetary value of a specified amount of nickel or copper. This issue enab
Inco to get out of its financial difficulties in 1984.
Cominco Ltd. of Canada also raised US$54 million in 1981 by issuing preferred-share and
commodity-indexed warrants (CIS). Holders of the CIS had the right to exchange each warra
or before August 1992 for a number of common shares of the corporation, based on the av
market price of zinc or copper and the market value of common stocks on the exercise dat
The largest producer of copper in the United States, Magna, issued copper-indexed notes in
The notes matured in 1998 and linked the interest payments to the price of copper. The int
rates ranged between 21 per cent per annum at average copper prices of US$2 per pound
6
below.
s. The
il.
apua
g of
rices
tors to
ade it
ld
e
r LDCs
es
tin,
umer
heme
tates
quotas
e.
er
the
ices
above, and 12 per cent per annum at average copper prices of 80 cents (US) per pound and
The indexation of the interest payments to copper prices enabled Magna to reorganize its
liabilities and therefore to control the risk to, and the net worth of, the company.
Commodity-indexed bonds have also been used in the LDCs to finance development project
government of Malaysia accepted a loan from Citibank that was indexed to prices of palm o
Similarly, Metallgesellschaft used copper-indexed financing to invest in the copper belt of P
New Guinea.
3. Ways to Protect Export Commodities from Price Volatility
For years, LDCs have been faced with colossal foreign debt. The retirement and/or servicin
this debt has been a major problem for LDCs and their creditors due to the volatility of the p
of export commodities and hence their export revenues. The crisis created by these debt
“overhangs” has drawn academics and practitioners to research ways and means for credi
receive, if not the principal, at least the interest payments on the debt. The crisis has also m
difficult for LDCs to obtain new loans.
The difficulty that LDCs face in meeting their debt obligations would be reduced if they cou
embark on measures that would protect their export commodities from price volatilities. On
measure suggested in the literature is that LDCs adopt hedging strategies. Whereas some
researchers suggest the use of futures markets by these countries, other researchers call fo
to shift the risk that their commodity prices face to the financial markets. Fall (1986) describ
three methods LDCs use to hedge against the risk their export commodity prices face:
international commodity agreements (ICAs), the futures markets, and countertrade.
3.1 International commodity agreements
ICAs, which cover commodities such as cocoa, coffee, natural rubber, olive oil, sugar, and
have been in effect for a number of years. Through these agreements, the LDCs and cons
countries sign a pact that seeks to stabilize world prices of commodities. This stabilization sc
is carried out to attract importers and satisfy the interest of producing countries. Fall (1986) s
that producing countries prefer price-supporting systems that are achieved through export
or buffer stocks. ICAs allow prices of commodities to fluctuate freely within an agreed rang
Whenever prices fall through the floor, export quotas are applied or the buffer-stock manag
enters the market and purchases sufficient amounts of the commodity. Either action raises
price of the commodity to fall within the predetermined range. On the other hand, should pr
7
the
faced
d, it
rice
be
ty is
fore
t or
all
s can
ge
ctions
ficulty
s
tput.
this
heme
-
go through the ceiling, the export quotas are relaxed or the buffer-stock manager sells the
commodity in the spot market to drive the price down to within the range.
ICAs have been fraught with three problems. First, there is asymmetry in the incentives of
importers and the producing LDCs in entering into the agreement. Whereas the consumers
(importers) are mainly concerned that higher prices will reduce their purchasing power of
imports, the producers are concerned with low prices. Second, the buffer-stock manager is
with limited funds to purchase the commodity whenever the price falls through the floor. Thir
is extremely difficult to get all the signatories to ICAs to abide by the quotas whenever the p
falls through the floor.
3.2 Futures market
By entering into the futures market, LDCs can lock in the price at which the commodity will
sold in the future. However, futures contracts have their limitations. First, their term to maturi
about two years. Second, regulations at the futures exchanges restrict investors (and there
LDCs) from taking huge positions in the markets, to prevent them from cornering the marke
manipulating prices. These limitations suggest that LDCs may not be in a position to hedge
their exports through the futures market.
3.3 Countertrade
Countertrade, defined as a financing scheme in which settlements are made in the form of
physical goods instead of money, can take three forms. The first is the barter system: LDC
have bilateral or multilateral arrangements with developed economies in which they exchan
their export commodities for other goods produced by the developed countries. The transa
can take place instantaneously or within a year. The weakness of the barter system is its dif
in matching the interests of participating parties. This problem is known as the “double
coincidence of wants.”
The second form of countertrade strategy is “buyback arrangements.” In this strategy, LDC
import production facilities and agree to deliver at some future date a specified amount of ou
These arrangements most often involve the financing of processing plants in LDCs. Under
scheme, LDCs are able to lock in the present the future earnings of output. Although the sc
does not insulate producer countries from the risk of volatile commodity prices, it is project
specific.
8
ly
nder
e
milar
to
ajor
. The
of the
orea.
ram of
cial
uctural
ir
o
f the
tment
use
eded to
ic
teral
alled
ank
ges at
ified
The third form of countertrade strategy is known as a “counterpurchase agreement.” In this
strategy, an LDC imports certain commodities from a developed country and simultaneous
commits itself to export to that country a specific amount of commodities at an agreed date. U
this arrangement, LDCs are protected against export-commodity price risk. Furthermore, th
transactions made under this arrangement are similar to the importing LDC entering into a
mixture of spot and forward contracts with the developed economies. Hence, LDCs enjoy si
advantages offered by forward contracts.
3.4 The Baker plan
Despite the availability of the above-noted hedging schemes, an enormous debt continues
“overhang” LDCs, which has prompted a call for debt reorganization. The United States, a m
creditor of LDCs, has tried to use two different plans to help relieve and solve the debt crisis
first plan, known as the “Baker plan,” was proposed by Mr. James Baker, the U.S. Secretary
Treasury, at the October 1985 annual meeting of the IMF and World Bank in Seoul, South K
The Baker plan consisted of three parts and aimed to solve the debt problem through a prog
sustained growth for the economies of LDCs. In the first part of the plan, international finan
institutions encouraged debtor countries to develop comprehensive macroeconomic and str
policies that would enhance their growth, adjust their balance of payments, and reduce the
inflation rates. The second part of the plan called on the international financial institutions t
continue lending to LDCs that embarked on structural adjustment policies. In the third part o
plan, the private banks increased their lending in support of comprehensive economic adjus
programs.
It was Secretary Baker’s aim that, by implementing his plan, LDCs would be encouraged to
austere economic measures to curb inflation, and encouraged to produce trade surpluses ne
service foreign debt. The structural adjustment and new foreign lending would spur econom
growth for the LDCs and consequently reduce their debt load.
The Baker plan, however, failed to achieve its purpose, because the private and the multila
banks did not increase their lending, and the LDCs, for political reasons, were not able to
implement the structural adjustment policies. As a result of this failure, the United States, in
March 1985, implemented a scheme known as the “Brady plan.”
The plan, which was announced by Mr. Nicholas Brady, the U.S. Secretary of the Treasury, c
for the forgiveness of part of the LDC’s debt. It also proposed that the IMF and the World B
extend credit to debt-burdened nations so that they could collateralize debt-for-bond exchan
discounts and cash buybacks of debt, and ameliorate the interest payments on new or mod
9
e
Cs.
C that
s real
e
ation
in their
eign
oduce
hods
debt.
rgue
the
n in
erse
calls
argue
ty of
uld
erious
te
n
debt contracts. Kenen (1990) notes that, in accordance with the Brady plan, the IMF and th
World Bank extended new credit to Mexico, Costa Rica, and the Philippines.
3.5 Research on the policy of debt relief for LDCs
The Baker and the Brady plans led to academic research on the policy of debt relief for LD
Advocates of debt relief, such as Krugman (1989), suggest that reducing the debt of an LD
has a debt overhang could increase that country’s economic efficiency and consequently it
income, which would in turn lead to a reduction in its default risk. Kenen (1990) supports th
position of Krugman (1989) and Sachs (1988) by arguing that a country with a large debt
overhang suffers in two ways from a fall in economic efficiency. First, the high debt-service
payments made by debt-laden countries require high tax rates that discourage capital form
and the repatriation of capital. Second, since governments of heavily indebted LDCs are
responsible for making the debt-service payments and, therefore, those payments appear
budgets, they may not institute a devaluation policy that could be required to improve their for
reserve positions and consequently the debt crises.
Dornbusch (1988) notes that governments of LDCs use inefficient economic methods to pr
the trade surpluses needed to service their foreign debt. One reason for the inefficient met
may be the fact that devaluation increases the domestic-currency cost of servicing foreign
The higher cost raises the budget deficit and the growth rate of the money supply, and
consequently the inflation rate rises.
Other economists, such as Krugman (1988 and 1999), have used the debt Laffer curve to a
when debt forgiveness would be beneficial to LDCs. Krugman asserts that, if the LDC is on
correct (inclining) side of the debt Laffer curve, then debt forgiveness will lead to a reductio
the market value of outstanding debt, and therefore will be detrimental to creditors. The rev
holds true when the debtor country is on the wrong (declining) side of the Laffer curve. This
for the debtor country’s position on the Laffer curve to be determined before a decision on
forgiveness is made.
Froot, Scharfstein, and Stein (1989) point out the moral-hazard effect of forgiveness. They
that the amount of relief required to induce investment in the LDCs may depend on a varie
factors, some of which may be known only by the borrowing country. A borrowing country wo
know the level of austere economic measures it can impose on its citizens without causing s
disruptions. Hence, in negotiating for debt relief, the country might conceal part of the priva
information it has on its citizens in order to receive more relief. Froot, Scharfstein, and Stei
10
ure
DCs’
ure
up a
se to
r
e
icates
called
debt
Cs
hat
ed
when
years,
umber
st, by
ds
eturn
ked
lace
as the
se of
or
f the
o the
d in
believe that these problems could be resolved if the forgiven countries would index their fut
debt-service payment to commodity prices.
The failure of the Baker and Brady plans has led to recent calls for better management of L
debt. Krueger (2003) catalogues how some developing countries are attempting to restruct
their debts and the potential challenges they face. Caballero (2003) calls on the IMF to set
Contingent-Markets Department and a Crisis Department. Caballero’s proposal, which is clo
that of Williamson (2000), calls for the Contingent-Markets Department to be responsible fo
identifying a country’s sources of capital-inflow volatility that are potentially contractible. Th
Crises Department would be responsible for handling non-contractible shocks, such as
unexpected events and blunders. Through the creation of these departments, Caballero ind
that the debt of the emerging countries would be better managed. Other experts have also
on the IMF to be focused, transparent, predictable, and quick to intervene in countries facing
crisis (Meltzer 2000, Williamson 2000, and Fischer 2002).
Avenues available to LDCs for hedging against fluctuations in the prices of their export
commodities are fraught with great difficulties for them. It is therefore important that the LD
find alternative means to deal with their growing external debt crisis. This paper proposes t
LDCs consider raising capital on the financial markets through the issue of commodity-link
bonds.
The use of commodity-indexed bonds, as O’Hara (1984) notes, dates as far back as 1863,
the Confederate States of America (CSA) issued bonds payable in bales of cotton. In recent
several commodity-indexed bonds have been issued on the financial markets. There are a n
of economic reasons why LDCs should be encouraged to issue commodity-linked bonds. Fir
issuing commodity-indexed bonds, governments and corporations that need investment fun
could share the appreciating market value of underlying commodities with bondholders in r
for a lower coupon rate. Furthermore, as Budd (1983) argues, the issuing of commodity-lin
bonds offers an opportunity for commodity-producing issuers and international commodity
organizations to borrow at below-market interest rates. Through this process, LDCs could p
themselves in an advantageous position by being linked to the international markets, such
U.S. commodity markets and Eurobond markets.
Second, countries with a higher chance of defaulting on the final payment of a bond, becau
serious balance-of-payment problems, could minimize the probability of default by asking f
higher coupon payments during the life of the bond, in exchange for paying the minimum o
bond’s face value and the monetary value of a pre-specified unit of the commodity indexed t
bond. The default probability is reduced because the contractual debt payments are reduce
11
st
faced
s and
in the
who
arket
s on
ly to
re,
s,
dity-
e year
an
rs of
ents
legal
bond
ies in
rer of
e
an
us
LDC
ty of
suers
precisely those circumstances when balance-of-payments problems occur. Also, under this
arrangement the maximum the issuer would pay on the maturity date is the face value.
Third, the LDCs could, through the issue of bonds linked to their main exports, hedge again
fluctuations in their export earnings. Myers and Thompson (1989) note that the debt crisis
by the LDCs are due to a fall in export revenues and a simultaneous rise in world interest rate
debt-service payments. Myers and Thompson argue that, if the LDCs’ debt had been issued
form of commodity-linked bonds, then the debt-service payment of the LDCs would have
declined along with export prices (or export revenues), thus lightening their debt load. Those
oppose LDCs issuing commodity-linked bonds suggest that LDCs should use the futures m
to control for commodity price risk. Regulators of the futures markets, however, impose limit
the movements of the futures price in a single day. Thus, futures prices cannot move quick
accommodate new information. Such limits are not in place for commodity options; therefo
commodity-linked bonds, which are a combination of straight bonds and commodity option
would react to new information to form the equilibrium price. Another advantage of commo
linked bonds over futures contracts is that futures contracts have a maturity of less than on
and exist for a limited number of commodities. By issuing commodity-linked bonds, LDCs c
have longer-term maturity and also index the bonds to any commodity of their choice.
Fourth, the issuance of commodity-linked bonds minimizes the default risk faced by financie
LDC loans. A way still must be found, however, to reach the necessary collateral arrangem
between LDCs and the developed nations that are major holders of the bond. One way is a
contract between the LDCs and investing nations such that holders of a commodity-linked
are empowered to seize any proceeds from the LDCs’ exports in any of the signatory countr
the case of default. The drawback is that such a contract is not enforceable, and enormous
transactions costs would have to be incurred to settle a dispute between an LDC and a bea
the bond. Kletzer and Wright (2000), however, demonstrate that, in the presence of credibl
punishment threats, sovereign borrowers would always choose to renegotiate an existing lo
contract rather than default.
Fifth, the use of commodity-linked bonds for external financing would minimize the enormo
transactions costs that would be incurred if LDCs were to dynamically hedge their export
revenues with futures contracts. In this paper, the model of Myers and Thompson (1989) is
extended to determine the optimal proportion of total external debt that must be issued by an
in the form of commodity-linked bonds.
Sixth, in a world of inflation, and given the general uncertainties in the markets, the availabili
the commodity, indexed to the bonds, greatly reduces the default risk of the bonds. Hence, is
12
rve
he
the
ld
f
e
re
ort
inked
n the
tants.
ge tar-
of the bonds must maintain a threshold level of inventory similar to what banks hold as rese
requirements. Moreover, issuers of the bonds who do not have the commodity must back t
bonds with a long position in the forward or futures contracts, whose maturity is timed with
redemption date of the bonds.
4. A Model of Optimal External Debt Allocation
In this section, the framework of portfolio theory is applied to derive simple rules LDCs cou
follow in allocating debt instruments and their level of imports.3 Besides the usual assumptions o
no taxes, continuous trading, and zero transactions costs made in the financial literature, th
following assumptions are made: the LDC has a small open economy; all prices of assets a
denominated in U.S. dollars; all external debt is issued by the government; there are no sh
sales, because a country cannot sell short its own debt; two sources of foreign finances are
available to the government (the issue of conventional bonds and the issue of commodity-l
bonds); there is only one perishable and divisible imported good; and the rates of change i
price of the export commodity and the Libor rate follow a stochastic Brownian motion.
4.1 Conventional debt
The process followed by the price of the export commodity is postulated as:
, (1)
whereαp is the instantaneous average return of holding one unit of the export commodity,σp is the
instantaneous standard deviation of the rate of change of the commodity price, anddzp has a
standard normal distribution with a mean of zero and a variance ofdt. Note thatαp andσp may be
functions ofP andt. For the purpose of this exercise, however, they are assumed to be cons
The Libor rate is assumed to follow a mean-reversion stochastic process of the form:
. (2)
The parameters are all constants. The Libor rate tends to be pulled towards the avera
get,θ. σr is the instantaneous standard deviation for the rate of change of the Libor rate, anddzr is
3. See Merton (1971, 1973) on the methodology followed herein.
PdP------ αp t σp zpd+d=
rd κ θ r–( )dt σrdzr+=
13
n that
g a
normally distributed with a mean of zero and a variance ofdt. Also,dzp anddzr have an instanta-
neous correlation ofρprdt.
Let the price of conventional debt,Q(r, t), be dependent on the Libor rate. By applying
Ito’s Lemma, the rate of change of the price,Q(r, t), is given as:
, (3)
where,
, (4)
and
. (5)
Standard arbitrage arguments can be advanced to show that the partial differential equatio
governs the pricing of the conventional debt with a coupon payment ofc is given as:
, (6)
whereλ(r) is the market price of risk attached to all financial assets whose underlying state
variable is the Libor rate. Also,r is the instantaneous riskless rate of interest. For a given
boundary condition, a closed-form solution for equation (6) cannot be determined. Assumin
face value ofQ0, no coupon payments, and a constant market price of interest rate risk (orλ(r) =
λ), Vasicek (1977) shows that the price of the conventional debt satisfies:
, (7)
whereτ is the time left to maturity, and
. (8)
dQQ
------- αqdt σqdzr+=
αq
κ θ r–( )Qr 0.5σr2Qrr Qt–+
Q-------------------------------------------------------------------=
σq
σrQr
Q------------=
κ θ r–( ) σr λ r( )–[ ]Qr 0.5σr2Qrr Qt– rQ– c+ + 0=
Q r τ,( ) Q01κ--- 1 e
κτ––( ) A r–( ) τA–
σr2
4κ3--------- 1 e
κτ––( )
2–exp=
A θσr λκ
---------12---
σr2
κ2------–+=
14
l debt
the
and
ation
Alternatively, if we assume a constant interest rate, then the market value of the conventiona
will be:
. (9)
Based on either equation (7) or (9), the main driver for the conventional debt is found to be
level of the interest rate.
4.2 Commodity-linked bond
Consider a commodity-linked bond, the value of which is solely a function of the Libor rate
the price of the export commodity. LetH(r, P, t) be the price of a commodity-linked bond.
Applying Ito’s lemma, the rate of change of the commodity-linked bond is obtained as:
, (10)
where
(11)
and
, (12)
. (13)
The application of standard arbitrage arguments yields the partial differential equation that
governs the valuation of the commodity-linked bond, which is of the form4:
4. See Schwartz (1982), Atta-Mensah (1992), or Miura and Yamauchi (1998) for expanded valumodels of commodity-linked bonds.
Q P τ,( ) cr-- 1 e
rτ––( ) Q0e
rτ–+=
dHH
------- αhdt ψrdzr ψpdzp+ +=
αh κ θ r–( ) σr λ r( )–[ ]Hr αpPHp 0.5σp2P
2H pp 0.5σr
2Hrr
ρprσpσrPHpr Ht–
+ + +
+
{} H⁄ ,
=
ψp
σpPHp
H------------------=
ψr
σr Hr
H------------=
15
14)
a
m of
:
of a
(14)
In equation (14),cB is the coupon rate of the commodity-linked bond. Furthermore, equation (
is restricted by the following conditions:
H(r, 0) = 0 ∀ r, (15)
H(r, ∞) = Q(r, t) ∀ r, (16)
H(∞, P) = 0 ∀ P. (17)
4.2.1 The value of the commodity-linked bond
The price of the commodity-linked bond is shown by the solution of equation (14) subject to
boundary condition. As stated earlier, a commodity-linked bond is indexed to an underlying
commodity. Assume that the promised payment on the bond at maturity is set at the maximu
the face value of the bond (F) and the monetary value of a pre-specified unit of the referenced
commodity. Letγ be the pre-specified unit of the commodity referenced to the bond, andHc(⋅) the
value of this particular bond; then, notationally, the final payment of the bond is of the form
Hc(P, r, 0) = Max[F, γP], (18)
or,
Hc(P, r, 0) = F + γMax[0, P - F/γ]. (19)
Equation (19) implies that the promised payment of the bond is equivalent to the face value
bond (F) for sure, plusγ amounts of a call option, which gives the bearer an option to buy the
reference commodity bundle at a specified exercise price,F/γ.
On the other hand, to minimize default risk, the borrower could have an option to pay the
minimum of the face value and the value of the reference amount of the commodity at the
maturity date. In that case, the terminal value of the bond would be:
Hp(P, r, 0) = Min[F, γP], (20)
or,
κ θ r–( ) σr λ r( )–[ ]Hr 0.5σp2P
2H pp 0.5σr
2Hrr ρprσpσrPHpr+ + +
rPH p Ht– rH– cB
+ + 0= .
16
e,
alue,
he
r
hich
ked
Hp(P, r, 0) = F -γMax[0, F/γ − P]. (21)
Equation (21) indicates that a commodity-linked bond that pays the minimum of the face valuF,
and the monetary value of a pre-specified unit of a commodity is similar to a bond of face v
F, and a short position onγ amounts of a put option, which gives the bearer an option to sell t
reference commodity bundle at a specified exercise price,F/γ.
A closed-form solution of equation (14), subject to the boundary conditions, equation (19) o
equation (21), is not a trivial exercise. Hence, for expositional reasons, consider a case in w
the interest rate is constant. For simplicity and without loss of generality, also assume thatγ is
equal to unity. With these assumptions, the differential equation for pricing the commodity-lin
bond and boundary conditions simplifies to:
, (22)
and
Hc(P, 0) =F + Max[0, P - F], (23)
Hp(P, 0) =F - Max[0, F - P]. (24)
The solution of equation (22) subject to (23) is given as:
, (25)
wherecc is the coupon payment,L(P, F,τ) the Black-Scholes (1973) formula for valuing a call
option onP with exercise priceF, andτ the time left to maturity:
, (26)
where
, (27)
, (28)
andN(.) is the cumulative normal-distribution function.
0.5σp2P
2H pp rPH p Ht– rH– c
B+ 0=+
Hc
P τ,( ) cc
r---- 1 e
rτ––( ) Fe
rτ–L P F τ, ,( )+ +=
L Q F τ, ,( ) PN d1( ) Ferτ–
N d2( )–=
d1
P F⁄( ) r12---σp
2+
τ+log
σp τ----------------------------------------------------------=
d2 d1 σp τ–=
17
ation
onds
uld
of
On the other hand, the value of the bond could be the solution of equation (22) subject to equ
(24):
, (29)
wherecp is the coupon payment,Q(P, F,τ) the Black-Scholes (1973) formula for valuing a put
option onP with exercise priceF, andτ the time left to maturity:
, (30)
where
, (31)
, (32)
andN(.) is the cumulative normal-distribution function.
4.2.2 Commodity price and the value of the commodity-linked bond
Because the primary focus of this paper is to argue that LDCs could, through the issue of b
linked to their main exports, hedge against the fluctuations in their export earnings, one wo
expect the value of debt issued in the form of commodity-linked bonds to fall with the falling
prices of (or revenues from) exports.
Proposition 1: The value of the commodity-linked bond increases monotonically as the price
the commodity indexed to the bond increases.
Proof:
Differentiating equation (25) with respect toP:
, (33)
Hp
P τ,( ) cp
r----- 1 e
rτ––( ) Fe
rτ–Q P F τ, ,( )–+=
Q P F τ, ,( ) Ferτ–
N a1( ) PN a2( )–=
a1
F P⁄( ) r–12---σp
2+
τ+log
σp τ--------------------------------------------------------------=
a2 x1 σp τ–=
Hc∂
P∂--------- N d1( ) 1
σp τ-------------N′ d1( )+
Ferτ–
Pσp τ-----------------N′ d2( )–=
18
a
to
y
ey
but,
, (34)
thus,
. (35)
Substitute equation (34) in the last part of equation (35):
, (36)
which simplifies into:
. (37)
Hence:
. (38)
Alternatively, differentiating equation (29) with respect toP also yields:
. (39)
Remarks: The first type of commodity-linked bond is equivalent to a portfolio that consists of
discount bond with a face value ofF and a European call option on the commodity referenced
the bond with an exercise price ofF. An explanation for Proposition 1 is that, as the commodit
price increases, the probability that the call contained in the portfolio will end up in the mon
increases, which appreciates the value of the commodity-linked bond.
N′ x( ) 1
2π----------e
1 2⁄( )x2–=
Hc∂
P∂--------- N d1( ) 1
σp 2πτ-------------------- e
1 2⁄( )d12–
eP F⁄( ) r+log– τ 1 2⁄( )d2
2––+=
Hc∂
P∂--------- N d1( ) 1
σp 2πτ-------------------- e
1 2⁄( )d12–
ed1σp τ 1 2⁄( )σp
2τ 1 2⁄( ) d1 σp τ–( )2–+
–+=
Hc∂
P∂--------- N d1( ) 1
σp 2πτ-------------------- e
1 2⁄( )d12–
e1 2⁄( )d1
2––+=
Hc∂
P∂--------- N d1( ) 0≥=
Hp∂
P∂---------- N a2( ) 0≥=
19
ount
ption:
rice.
main
t
f they
e
.
d put
curs
ice
ince its
nd a
o an
duces
The second type of commodity-linked bond is equivalent to a portfolio that consists of a disc
bond with a face value ofF and a short position on a European put option on the commodity
referenced to the bond with an exercise price ofF. The value of the commodity-linked bond rises
with the increase in the price of the referenced commodity, because of the value of the put o
the chances of the put option finishing out of the money rises with the rise in the commodity p
The two results clearly show that, if LDCs had issued debt contracts that were tied to their
export commodities, then their debt load would have declined along with plummeting expor
prices (or export revenues). LDCs could therefore have prevented their current debt crisis i
had issued commodity-linked bonds.
Proposition 2: An LDC that has a volatile commodity price can minimize its debt burden by
issuing bonds that pay holders, on maturity, the lesser of the face value of the bond and th
monetary value of a pre-specified unit of a commodity, rather than the greater of these two
Proof:
Differentiating equation (25) with respect toσp and simplifying yields:
. (40)
Differentiating equation (29) with respect toσp yields:
. (41)
Remarks: Equations (40) and (41) show that a commodity-linked bond that has an embedde
option falls in value when the volatility of the commodity price rises, whereas the opposite oc
with a bond that has an embedded call option. The increased volatility of the commodity pr
increases its value option attached to the bond, because a put call has no downside risk, s
value is zero irrespective of how far it finishes out of the money. Hence, an increase in the
volatility of the commodity price increases the chances that the put option will expire in the
money. Given that the commodity-linked bond of this type is equivalent to a regular bond a
short position on a put option, the value of the bond falls with a rise in the volatility of the
commodity price. In other words, the heightened volatility of export commodity prices leads t
increase in the expected export revenue, and, with the debt burden falling with it, greatly re
the chance of an LDC defaulting on the bond.
Hc∂
σp∂--------- τPN′ d1( ) 0≥=
Hp∂
σp∂---------- τPN′ a2( )( )– 0≤=
20
tility
h the
e is
s its
n a
e
a
than
f the
or the
ose
t of a
e two
arket
t:
Alternatively, if the LDC was to issue a bond with an embedded call, then the rise in the vola
of the commodity price would increase the value of the bond. The value of the call rises wit
volatility of the commodity price, because there is no downside risk to the call, since its valu
zero irrespective of how far it finishes out of the money. An increase inσp, therefore, increases the
chances that the call option will expire in the money. The implication is that an LDC increase
debt burden when it issues commodity-linked bonds that are embedded with call options o
commodity price, because the value of the bond rises with the increase in the volatility of th
commodity price.
Proposition 3: In an environment where interest rates are not stochastic, the coupon rate for
conventional debt with an identical face value as a commodity-linked bond is generally less
the coupon rate for a commodity-linked bond that pays holders, on maturity, the minimum o
face value and the monetary value of a pre-specified unit of a commodity. The coupon rate f
conventional bond is, however, greater than its counterpart for a commodity-linked bond wh
terminal payoff is the greater of the face value and the monetary value of a pre-specified uni
commodity.
Proof:
Given their identical face values, an investor on the margin would be indifferent between th
types of commodity-linked bonds and a conventional bond, which implies that the current m
values of the two instruments must be the same. Using equations (9) and (22), and settingQ0 to F,
we have:
(42)
which implies that
. (43)
But , because there is no downward risk for an option. It therefore follows tha
. (44)
Similarly,
cr-- 1 e
rτ––( ) Fe
rτ–+
cc
r---- 1 e
rτ––( ) Fe
rτ–
L P F τ, ,( )
+ +
cp
r----- 1 e
rτ––( ) Fe
rτ–Q P F τ, ,( ),–+
=
=
cr-- 1 e
rτ––( ) c
c
r---- 1 e
rτ––( )– L P F τ, ,( )=
L P F τ, ,( ) 0≥
c cc
0≥–
21
ked
he
for
hat
of the
-
dity
ity
odity-
-
ge for
hat the
. (45)
Given that and ,
. (46)
Lastly,
, (47)
or
. (48)
Putting equations (44), (46), and (48) together, we have:
. (49)
Remarks: Proposition 3 strengthens the economic rationale for the issue of a commodity-lin
bond. It demonstrates that LDCs or corporations in need of investment funds could share t
appreciation of the market value of the underlying commodity with the bondholders, in return
a lower coupon rate. In this case, LDCs would benefit by issuing commodity-linked bonds t
pay, on maturity, the greater of the face value or the monetary value of a pre-specified unit
underlying commodity. This supports Budd (1983), who argues that the issue of commodity
linked bonds offers an opportunity for commodity-producing issuers and international commo
organizations to borrow at below-market interest rates.
On the other hand, an LDC could share the depreciation of the market value of its commod
price with bondholders in exchange for higher coupon rates. The LDC would issue a comm
linked bond whose final payoff is the lesser of the face value or the monetary value of a pre
specified unit of the underlying commodity. The issuance of such bonds would act as a hed
an LDC during times when the commodity price experiences a collapse (Caballero 2003).
4.3 Net foreign debt
Without external financing, the value of imports must equal the value of exports, so that the
current account is in balance each period. The assumption made in this paper, however, is t
cp
r----- 1 e
rτ––( ) c
c
r---- 1 e
rτ––( )– L P F τ, ,( ) Q P F τ, ,( ) 0≥+=
L P F τ, ,( ) 0≥ Q P F τ, ,( ) 0≥
cp
cc
0≥–
cp
r----- 1 e
rτ––( ) c
r-- 1 e
rτ––( )– Q P F τ, ,( ) 0≥=
cp
c 0≥–
cp
c cc≤ ≤
22
t of
n
,
bond
total
bill is
government of the LDC has access to two sources of external financing: one is to issue
conventional debt and the other is to issue a commodity-linked bond.
Let D(t) = be the quantity of conventional debt outstanding to the governmen
the LDC.5 The new quantity of debt issued in each period is, therefore, =dD/dt. Similarly,
the total quantity of commodity-linked bonds outstanding isB(t) = . The quantity of
new commodity-linked bonds issued is =dB/dt. Furthermore, assume that both the
conventional debt and the commodity-linked bond are of the console type. Also, the coupo
payments to bearers of conventional debt and the commodity-linked bonds are, respectivelyc and
cB. Hence, in each period, the contributions of the conventional debt and commodity-linked
to the net foreign debt of the government are, respectively,Q dt -Dc andH dt - BcB.
If x is the fixed rate of commodities exported andm(t) is the rate of imports consumed, then, in
every instant, imports must be financed by the sum of export revenue and the value of new
debt less the total coupon payments. In other words, the government’s instantaneous import
constrained by the following function:
. (50)
Let W be the value of the total external debt of the government of the LDC:
. (51)
The change inW is, therefore,
. (52)
But the import constraint of equation (50) shows that:
. (53)
Substituting equation (53) into equation (52),
. (54)
Defineω1 as the fraction of the total external debt held in conventional debt andω2 as the fraction
of external debt held in commodity-linked bonds:
5. D(t-1) is a conventional debt that matures int-1 periods.
D t 1–( ) τd0t∫
D t( )B t 1–( ) τd
0t∫
B t( )
D B
m t( )dt Pxdt QdD HdB Dcdt– BcBdt–+ +=
W QD HB+=
dW DdQ BdH+ QdD HdB++=
QdD HdB+ m t( ) Pxdt– Dcdt BcBdt+ +=
dW DdQ BdH m t( ) Pxdt– Dcdt BcBdt+ + + +=
23
and
port
tion of
s the
s
ω1 = QD/W and ω2 = HB/W.
Equation (54) then becomes:
. (55)
Note thatQ andH must satisfy equations (8) or (9) and (25) or (29). Substitute equations (3)
(10) into equation (55) and note thatω1 + ω2 = 1. Sinceω2 = 1 -ω1, the flow of the net external
debt is:
(56)
Equation (56) demonstrates that the value of the external debt of the LDC changes with the
market valuations of conventional bonds and commodity-linked bonds, the import bill, and ex
revenue. Shocks from interest rates and commodity prices, however, make the market valua
the debt very uncertain.
4.4 The government’s maximization problem
The government is faced with choosing in each period the level of imports,m, and the fractions of
total external debt,ω1 andω2, that must be held in conventional debt and commodity-linked
bonds. The government embarks on this portfolio and imports rule in a manner that maximize
expected value of a time-additive von Neumann-Morgenstern utility function. The problem i
formulated as:
, (57)
subject to equation (56) and
. (58)
dW ω1WdQQ
------- ω2WdHH
------- mdt Pxdt–ω1W
Q------------cdt
ω2W
H------------c
B+ + + +=
dW ω1W αq αh– c Q cB
H⁄–⁄+( ) m Px–+[=
W αh cB
H⁄+( )+ ]dt ω1W σq ψr–( ) Wψr+[ ]dzr+
1 ω1–( )+ Wψpdzp .
Max
m ω1,E0 e
βt–U m t( ) t,( )
0
∞
∫ dt
W 0( ) W0=
24
e
to:
hat
made.
Also, the utility functionU(⋅) is restricted to be concave inm (i.e.,Um > 0 andUmm< 0).E0 is the
expectations operator, conditional onW(0) = W0 being known.
Using dynamic programming techniques, aJ function can be defined as:
. (59)
Equation (59) is also constrained by equations (56) and (58). Equation (59) can therefore b
rewritten as,
. (60)
As shown in the appendix, the optimization problem that faces the government is reduced
, (61)
whereL, which is known as the Dynkin operator over the variablesW, P, and r, is defined in the
appendix. The first-order condition for a maximization problem is:
, (62)
(63)
Before finding the optimum proportions of commodity-linked bonds and conventional debt t
must be raised by the government externally, some comments on equation (62) should be
J W P r t, , ,( )Max
m ω1,E0 e
βt–U m t( ) t,( )
0
∞
∫ dt≡
J W t0( ) P r t0, , ,( )Max
m ω1,Et0
eβt–
U m t( ) t,( )dt J W t1( ) P r t1, , ,( )( )+
t0
t1
∫≡
Max
m ω1,Φ ω m W P r t, , ,;,( ) e
βt–U m t( ) t,( ) L J( )+=
φm eβt–
Um Jw+ 0= =
φω1JwW αq αh– c Q c
BH⁄–⁄+( )=
J+ wpWPσp ρpr σq ψr–( ) ψp–( ) JwrWσr σq ψr–( ) ρrpψp–( )+
0.5+ JwwW2
2( ω1 σq ψr–( )22ψr σq ψr–( )+
2ρprψp 1 2ω1–( ) σq ψr–( ) 2ρprψpψr– 2 1 ω1–( )ψp2 )–+ 0= .
25
is
re the
inal
ted
be
d
ot add
Equation (62) implies that the marginal utility of external debt to the government of an LDC
negative. The LDC, therefore, chooses an optimum level of imported goods at the point whe
sum of marginal utility derived from consuming imported goods and the marginal utility of
external debt is zero. In other words, LDCs will contract loans up to the point where the marg
disutility of total external debt is completely offset by the marginal utility derived from impor
goods.
4.5 Optimal allocation of external debt
Equation (63) is used to obtain the optimum proportions of the total external debt that must
held in conventional debt and commodity-linked bonds. Thus, rearranging equation (63) an
simplifying, the optimum weight of conventional debt is expressed as:
(64)
Without loss of generality, the last term of equation (64) could be dropped, because it does n
much to the discussion. The optimum proportion of external debt that is in the form of
commodity-linked bonds is given as:
. (65)
Thus,
ω1* Jw
W Jww---------------
αq αh– c Q cB
H⁄–⁄+
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------–=
JwpP
WJww---------------–
ρprσp σq ψr–( ) σpψp–
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------
Jwr
WJww---------------–
σr σq ψr–( ) σr ρprψp–
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------
+ψr σq ψr–( ) ψp ρprψr ψp–( )+
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------.
ω2*
1 ω1*
–=
26
te of
rnal
l debt
e total
(66)
Assume that the government of the LDC has a logarithmic utility function with a constant ra
time preferenceγ. Also, let the ratio of the government’s instantaneous import bill to the exte
debt beλ. Thus,λ = m/W. With these equations, we have:
, (67)
. (68)
Equations (62), (66), and (67) can be used to obtain an expression for theJ(⋅) value function:
, (69)
whereΓ(⋅) is a function of the underlying state variables in the economy other thanW.
Applying equation (68), the optimum proportions of the total external debt in the form of
conventional debt and commodity-linked bonds are expressed as:
, (70)
and
. (71)
From equations (69) and (70) it can be seen that the optimal proportions of the total externa
raised in commodity-linked bonds and conventional debt depend on the spread between th
ω2*
1Jw
WJww---------------
αq αh– c Q cB
H⁄–⁄+
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------+=
JwpP
WJww---------------+
ρprσp σq ψr–( ) σpψp–
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------
Jwr
WJww---------------+
σr σq ψr–( ) σr ρprψp–
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------.
U m t,( ) eγt–
m( )log=
m*
W P r t, , ,( ) λW=
J W P r t, , ,( ) 1 λ⁄( )e γt–W( ) Γ P r t, ,( )+log–=
ω1* αq c Q⁄+( ) αh c
BH⁄+( )–
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------=
ω2*
1αh c
BH⁄+( ) αq c Q⁄+( )–
σq ψr–( )22ρprψp σq ψr–( )– ψp
2+
---------------------------------------------------------------------------------------+=
27
e
that
f the
try
that
at an
d total
n
n
edge
ses as
d debt
for a
rice
.
value
e of
ave to
es of
of
er
and the
returns (capital gains and coupon payments) of both bonds, adjusted by the riskiness of th
portfolio.6 The results accord with the literature on capital asset pricing. It can also be seen
the proportions respond positively to the debt’s own total return and negatively to the return o
alternative debt instrument. Note thatω1 andω2 would have to be non-negative, because a coun
cannot sell short its own debts. As in Merton (1971), equation (70) provides a rule of thumb
could be followed by an LDC in its investment decisions. For example, the rule suggests th
LDC should hold a larger share of commodity-linked bonds in its external debt portfolio
whenever the variance of the portfolio is greater than twice the spread between the expecte
return of the conventional debt and the commodity-linked bond.
5. Conclusion
In this paper, it has been argued that the issue of commodity-linked bonds would provide a
opportunity for commodity-producing developing countries to tie their borrowing needs to a
endowed resource. By issuing bonds indexed to their main export commodity, LDCs could h
against fluctuations in their export earnings and at the same time lessen the probability of
defaulting on their external debt obligation.
Results reported in this paper indicate that the value of the commodity-linked bonds increa
the price of the commodity indexed to the bonds rises. This suggests that, if LDCs had issue
contracts that were tied to their main export commodities, then their debt loads would have
declined along with plummeting export prices (or export revenues). This paper has also
demonstrated that the coupon rate for a commodity-linked bond is less than its counterpart
conventional debt instrument, if LDCs share, on maturity, the appreciation in the commodity p
with the bearer. The issuance of such bonds offers an opportunity for commodity-producing
issuers and international commodity organizations to borrow at below-market interest rates
On the other hand, LDCs could issue a bond whose terminal payoff is the lesser of the face
and the monetary value of a pre-specified unit of a commodity. The coupon rate for this typ
bond would have to be larger than that for a conventional bond, because investors would h
be compensated for accepting the prescribed terminal payoff. The importance of these typ
bonds is that they act as a hedge for LDCs against plummeting commodity prices.
Finally, using portfolio theory, a simple rule was derived for an LDC to follow in its allocation
debt instruments and the level of imports. The rule suggests that an LDC should hold a larg
6. Riskiness is measured here as the correlation between the export price and the Libor rate,variances of the prices of the debt instruments.
28
bt
d total
y-
ese
cially
nts. To
milar
t back
ith
share of commodity-linked bonds in its external debt portfolio than that of a conventional de
whenever the variance of the portfolio is greater than twice the spread between the expecte
return of the conventional debt and the commodity-linked bond.
Like most economic models, there are limitations to this model. The viability of a commodit
linked bonds market cannot be guaranteed by simply letting risk-prone speculators issue th
bonds to risk-averse hedgers. Hence, the commodity-linked bond market must be commer
guided and participants must be major market markers, such as corporations and governme
reduce default risk, the issuers of the bonds must maintain a threshold level of inventory, si
to what banks hold as reserves. Furthermore, issuers that do not have the commodity mus
the bonds with a long position in the forward or futures contracts, whose maturity is timed w
the redemption date of the bonds.
29
BER
ment
el.”
to-
n-
References
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Fall, M. 1986.Commodity-Indexed Bonds. Unpublished Masters thesis, M.I.T Sloan School ofManagement, Cambridge.
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31
Appendix
A.1 The Dynkin Operator
Let and assume that the third partial derivative ofJ(⋅) is bounded. By applying
Taylor’s series theorem, the mean value theorem for integrals, and taking the limits as
equation (46) becomes:
(A1)
However, the net foreign debt constraint (equation (42)) and equations (1) and (2) give:
(A2)
(A3)
t t0 ∆t+=
∆t 0,→
J W t0( ) P r t0, , ,( )Max
m ω1,U m t( ) t,( ) E J W t0( ) P r t0, , ,( )( )+[≡
Jtdt JwE dW( ) JpE dP( ) JrE dr( ) JwpE dWdP( ) JwrE dWdr( )+ + + + + +
JrpE drdP( ) 0.5JppE dP( )20.5Jrr E dr( )2]+ + .
E dW( ) ω1W αq αh– c Q cB
H⁄–⁄+( ) m Px–+[=
W αh cB
H⁄+( ) ]dt+ ,
E dW( )2 ω12W
2 σq ψr–( )22ω1W
2ψr σq ψr–( ) W2ψr
2+ +[=
2+ ρprW2 ψp ω1 ω1
2–( ) σq ψr–( ) 1 ω1–( )ψpψr+( )
1 ω1–( )2W
2ψp2 ]dt .,
32
, (A4)
, (A5)
, (A6)
, (A7)
, (A8)
, (A9)
. (A10)
Substituting equations (A2) to (A10) into equation (A1), and noting that
, the continuous-time version of the Bellman-Dreyfus
fundamental optimality equation is obtained, which is of the form:
E dP( ) αpPdt=
E dP( )2 σp2P
2dt=
E dr( ) κ θ r–( )dt=
E dr( )2 σr2dt=
E dWdP( ) WPρprσp ω( 1 σq ψr–( ) ψr ) 1 ω1–( )WPσpψp]dt+ +[=
E dWdr( ) σr ω1W σq ψr–( ) Wψr+( ) 1 ω1–( )Wσr ρrpψp]dt+[=
E dPdr( ) ρprσpσrPdt=
E J W t0( ) P r t, , ,( )( ) J W t0( ) P r t, , ,( )≡
33
(A11)
In compact form, equation (A11) can be expressed as:
,
whereL is the Dynkin operator over the variablesW, P, and r. This operator is defined as:
0Max
m ω1,U m t( ) t,( ) Jt+[≡
Jw ω1W αq αh– c Q cB
H⁄–⁄+( ) m Px–+(
W αh cB
H⁄+( ) ) Jp αpP( ) Jr κ θ r–( )( )+ + +
Jwp WPρprσp ω( 1 σq ψr–( ) ψr ) 1 ω1–( )WPσpψp+ +( )+
0.5Jww ω( 12W
2 σq ψr–( )22ω1W
2ψr σq ψr–( ) W2ψr
2+ ++
2+ ρprW2 ψp ω1 ω1
2–( ) σq ψr–( ) 1 ω1–( )ψpψr+( ) 1 ω1–( )2
W2ψp
2 )+
0.5σp2PJpp 0.5σr
2Jrr ]+ + .
φ m D B W P r t, , ,;, ,( ) U m t( ) t,( ) L J( )+=
34
(A12)
L J( ) Jt Jw ω1W αq αh– 1 Q 1 H⁄( )Min 1 βP,( )–⁄+( ) m Px–+(+=
W αh 1 H⁄( )Min 1 βP,( )+( ) ) Jp αpP( ) Jr κ θ r–( )( )+ + +
Jwp WPρprσp ω( 1 σq ψr–( ) ψr ) 1 ω1–( )WPσpψp+ +( )+
0.5Jww ω( 12W
2 σq ψr–( )22ω1W
2ψr σq ψr–( ) W2ψr
2+ ++
2+ ρprW2 ψp ω1 ω1
2–( ) σq ψr–( ) 1 ω1–( )ψpψr+( ) 1 ω1–( )2
W2ψp
2 )+
0.5σp2PJpp 0.5σr
2Jr ]+ + .
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